ask in vs quest

ask in is not a noun while quest is a noun.

Word	Noun	Adjective	Verb	Adverb
ask in	No	No	Yes	No
quest	Yes	No	Yes	No

As verbs, quest is a hypernym of ask in; that is, quest is a word with a broader meaning than ask in:

Other hypernyms of ask in include ask for, bespeak, call for, request.

ask in (noun)	quest (noun)
	the act of searching for something
	a search for an alternative that meets cognitive criteria

ask in (verb)	quest (verb)
ask to enter	seek alms, as for religious purposes
	express the need or desire for
	bark with prolonged noises, of dogs
	search the trail of (game)
	make a search (for)